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This is the book that Daniel Tammet, bestselling author and mathematical savant, was born to write. In Tammet''s world, numbers are beautiful and mathematics illuminates our lives and minds. Using anecdotes and everyday examples, Tammet allows us to share his unique insights and delight in the way numbers, fractions and equations underpin all our lives. Inspired by the complexity of snowflakes, Anne Boleyn''s sixth finger or his mother''s unpredictable behaviour, Tammet explores questions such as why time seems to speed up as we age, whether there is such a thing as an average person and how we can make sense of those we love.
About the author Daniel Tammet is the critically acclaimed author of the worldwide bestselling memoir, Born on a Blue Day, and the international bestseller Embracing the Wide Sky Tammet's exceptional abilities in mathematics and linguistics are combined with a unique capacity to communicate what it's like to be a savant His idiosyncratic world view gives us new perspectives on the universal questions of what it is to be human and how we make meaning in our lives Tammet was born in London in 1979, the eldest of nine children He lives in Paris Thinking in Numbers Daniel Tammet www.hodder.co.uk First published in Great Britain in 2012 by Hodder & Stoughton An Hachette UK company Copyright © Daniel Tammet 2012 The right of Daniel Tammet to be identified as the Author of the Work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means without the prior written permission of the publisher, nor be otherwise circulated in any form of binding or cover other than that in which it is published and without a similar condition being imposed on the subsequent purchaser A CIP catalogue record for this title is available from the British Library ISBN 978 444 73742 Extract from The Lottery Ticket by Anton Chekhov; Extracts from Lolita by Vladamir Nabokov © Vladamir Nabokov, published by Orion Books is used by permission; Extract of interview with Vladamir Nabokov was taken from the BBC programme, Bookstand and is used with permission; Extracts by Julio Cortazar from Hopscotch, © Julio Cortazar, published by Random House New York; Quote from The Master’s Eye translated by Jean de la Fontaine; Quote from Under the Glacier by Halldor Laxness, © Halldor Laxness, published by Vintage Books, an imprint of Random House New York Every reasonable effort has been made to acknowledge the ownership of the copyrighted material included in this book Any errors that may have occurred are inadvertent, and will be corrected in subsequent editions provided notification is sent to the author and publisher Hodder & Stoughton Ltd 338 Euston Road London NW1 3BH www.hodder.co.uk ‘To see everything, the Master’s eye is best of all, As for me, I would add, so is the Lover’s eye.’ Caius Julius Phaedrus ‘Like all great rationalists you believed in things that were twice as incredible as theology.’ Halldór Laxness, Under the Glacier ‘Chess is life.’ Bobby Fischer Contents Acknowledgements Preface Family Values Eternity in an Hour Counting to Four in Icelandic Proverbs and Times Tables Classroom Intuitions Shakespeare’s Zero Shapes of Speech On Big Numbers Snowman Invisible Cities Are We Alone? The Calendar of Omar Khayyam Counting by Elevens The Admirable Number Pi Einstein’s Equations A Novelist’s Calculus Book of Books Poetry of the Primes All Things Are Created Unequal A Model Mother Talking Chess Selves and Statistics The Cataract of Time Higher than Heaven The Art of Maths Acknowledgements I could not have written this book without the love and encouragement of my family and friends Special thanks to my partner, Jérôme Tabet To my parents, Jennifer and Kevin, my brothers Lee, Steven, Paul, and my sisters, Claire, Maria, Natasha, Anna-Marie, and Shelley Thanks also to Sigriður Kristinsdóttir and Hallgrimur Helgi Helgason, Laufey Bjarnadóttir and Torfi Magnússon, Valgerður Benediktsdóttir and Grímur Björnsson, for teaching me how to count like a Viking To my most loyal British readers Ian and Ana Williams, and Olly and Ash Jeffery (plus Mason and Crystal!) I am grateful to my literary agent Andrew Lownie; and to Rowena Webb and Helen Coyle, my editors Preface Every afternoon, seven summers ago, I sat at my kitchen table in the south of England and wrote a book Its name was Born On A Blue Day The keys on my computer registered hundreds of thousands of impressions Typing out the story of my formative years, I realised how many choices make up a single life Every sentence or paragraph confided some decision I or someone else – a parent, teacher or friend – had taken, or not taken Naturally I was my own first reader, and it is no exaggeration to say that in writing, then reading the book, the course of my life was inexorably changed The year before that summer, I had travelled to the Center for Brain Studies in California The neurologists there probed me with a battery of tests It took me back to early days in a London hospital when, surveying my brain for seizure activity, the doctors had fixed me up to an encephalogram machine Attached wires had streamed down and around my little head, until it resembled something hauled up out of the deep, like angler’s swag In America, these scientists wore tans and white smiles They gave me sums to solve, and long sequences of numbers to learn by heart Newer tools measured my pulse and my breathing as I thought I submitted to all these experiments with a burning curiosity; it felt exciting to learn the secret of my childhood My autobiography opens with their diagnosis My difference finally had a name Until then it had gone by a whole gamut of inventive aliases: painfully shy, hyper sensitive, cack-handed (in my father’s characteristically colourful words) According to the scientists, I had high-functioning autistic savant syndrome: the connections in my brain, since birth, had formed unusual circuits Back home in England I began to write, with their encouragement, producing pages that in the end found favour with a London editor To this day, readers both of the first book and of my second, Embracing the Wide Sky, continue to send me their messages They wonder how it must be to perceive words and numbers in different colours, shapes and textures They try to picture solving a sum in their mind using these multidimensional coloured shapes They seek the same beauty and emotion that I find in both a poem and a prime number What can I tell them? Imagine Close your eyes and imagine a space without limits, or the infinitesimal events that can stir up a country’s revolution Imagine how the perfect game of chess might start and end: a win for white, or black or a draw? Imagine numbers so vast that they exceed every atom in the universe, counting with eleven or twelve fingers instead of ten, reading a single book in an infinite number of ways Such imagination belongs to everyone It even possesses its own science: mathematics Ricardo Nemirovsky and Francesca Ferrara, who specialise in the study of mathematical cognition, write that, ‘Like literary fiction, mathematical imagination entertains pure possibilities.’ This is the distillation of what I take to be interesting and important about the way in which mathematics informs our imaginative life Often we are barely aware of it, but the play between numerical concepts saturates the way we experience the world This new book, a collection of twenty-five essays on the ‘maths of life’, entertains pure possibilities According to the definition offered by Nemirovsky and Ferrara, ‘pure’ here means something immune to prior experience or expectation The fact that we have never read an endless book, or counted to infinity (and beyond!) or made contact with an extraterrestrial civilisation (all subjects of essays in the book) should not prevent us from wondering: what if? Inevitably, my choice of subjects has been wholly personal and therefore eclectic There are some autobiographical elements but the emphasis throughout is outward looking Several of the pieces are biographical, prompted by imagining a young Shakespeare’s first arithmetic lessons in the zero – a new idea in sixteenth-century schools – or the calendar created for a Sultan by the poet and mathematician Omar Khayyam Others take the reader around the globe and back in time, with essays inspired by the snows of Quebec, sheep counting in Iceland and the debates of ancient Greece that facilitated the development of the Western mathematical imagination Literature adds a further dimension to the exploration of those pure possibilities As Nemirovsky and Ferrara suggest, there are numerous similarities in the patterns of thinking and creating shared by writers and mathematicians (two vocations often considered incomparable) In ‘The Poetry of the Primes’, for example, I explore the way in which certain poems and number theory coincide At the risk of disappointing fans of ‘mathematically-constructed’ novels, I admit this book has been written without once mentioning the name ‘Perec’ The following pages attest to the changes in my perspective over the seven years since that summer in southern England Travels through many countries in pursuit of my books as they go from language to language, accumulating accents, have contributed much to my understanding Exploring the many links between mathematics and fiction has been another spur Today, I live in the heart of Paris I write full-time Every day I sit at a table and ask myself: what if? Daniel Tammet Paris March 2012 Family Values In a smallish London suburb where nothing much ever happened, my family gradually became the talk of the town Throughout my teens, wherever I went, I would always hear the same question, ‘How many brothers and sisters you have?’ The answer, I understood, was already common knowledge It had passed into the town’s body of folklore, exchanged between the residents like a good yarn Ever patient, I would dutifully reply, ‘Five sisters, and three brothers.’ These few words never failed to elicit a visible reaction from the listener: brows would furrow, eyes would roll, lips would smile ‘Nine children!’ they would exclaim, as if they had never imagined that families could come in such sizes It was much the same story in school ‘J’ai une grande famille,’ was among the first phrases I learned to say in Monsieur Oiseau’s class From my fellow students, many of whom were single sons or daughters, the sight of us together attracted comments that ranged all the way from faint disdain to outright awe Our peculiar fame became such that for a time it outdid every other in the town: the onehanded grocer, the enormously obese Indian girl, a neighbour’s singing dog, all found themselves temporarily displaced in the local gossip Effaced as individuals, my brothers, sisters and I existed only in number The quality of our quantity became something we could not escape, it preceded us everywhere: even in French, whose adjectives almost always follow the noun (but not when it comes to une grande famille) With so many siblings to keep an eye on, it is perhaps little wonder that I developed a knack for numbers From my family I learned that numbers belong to life The majority of my maths came not from books but from regular observations and interactions day to day Numerical patterns, I realised, were the matter of our world To give an example, we nine children embodied the decimal system of numbers: zero (whenever we were all absent from a place) through to nine Our behaviour even bore some resemblance to the arithmetical: over angry words, we sometimes divided; shifting alliances between my brothers and sisters combined and recombined them into new equations We are, my brothers, sisters and I, in the language of mathematicians, a ‘set’ consisting of nine members A mathematician would write: S = {Daniel, Lee, Claire, Steven, Paul, Maria, Natasha, Anna, Shelley} Put another way, we belong to the category of things that people refer to when they use the number nine Other sets of this kind include the planets in our solar system (at least, until Pluto’s recent demotion to the status of a non-planet), the squares in a game of noughts and crosses, the players in a baseball team, the muses of Greek mythology and the Justices of the US Supreme Court With a little thought, it is possible to come up with others, including: {February, March, April, May, August, September, October, November, December} where S = the months of the year not beginning with the letter J her age exceeded the maximum figure of most mortality tables At 113 she became the oldest person in the world: la doyenne de l’humanité Even then, she did not die Her eyes dimmed, her joints stiffened, but she remained in good spirits Statisticians who had long estimated a maximal human life span of between 110 and 115 years were proven wrong Madame Calment became the first person in history to celebrate her one hundred and sixteenth birthday, her one hundred and seventeenth birthday, her one hundred and eighteenth birthday, her one hundred and nineteenth birthday In 1995, ‘Jeanne’ as people now called her, attained 120 years of age The nursing home, which until then had received few visitors, suddenly swelled with reporters from around the world Tiny, desiccated, she sat before a bank of cameras She does not smile, complained one of the photographers Ask her if she wants to go on living, demanded a journalist The nursing home director cupped her hand around the old woman’s right ear, as though playing an absurd game of Chinese whispers ‘The monsieur would like to know if you want to live a little longer?’ she shouted into the ear Yes Her notary was absent that day Long retired, nearing eighty, he was not well enough to attend Later that year, he died He had never set foot inside Madame Calment’s house Under the terms of the thirty-year-old contract, his widow continued to pay By the time Madame Calment eventually died two years later, aged 122, the notary and his family had paid nearly a million francs: many times the anticipated payout and twice what the property had been worth when they took on the arrangement From where, we might well ask ourselves, did this strange idea of an average cancer patient or an average nonagenarian originate? A Treatise on Man and the development of his faculties , introduced l’homme moyen (‘the average man’) to the nineteenth-century world ‘If an individual at any given epoch of society possessed all the qualities of the average man, he would represent all that is great, good, or beautiful.’ The author – a Belgian mathematician called Alphonse Quetelet – imagined how Nature created men He pictured Nature as a bowman who aimed continuously for the statistical centre The bull’seye would be a perfectly median man (or woman): a rational, temperate person free from any excess or deficiency Most individuals, according to this view, were Nature’s errant arrows They straggled, nearer or farther, around the mean The taller-than-average flaunted the shorter man’s missing inches; the drunkard lacked his part of the teetotaller’s (excessive) self-control; the hairy man wore the bald man’s follicles Quetelet therefore urged his readers to take a wide view of humanity Every person, after all, was ‘but a fraction of mankind’ Instead of interesting themselves with men and women, scientists should study the population as a whole ‘The greater the number of individuals observed, the more individual peculiarities, whether physical or moral, become effaced, and allow the general facts to predominate, by which society exists and is preserved.’ So the mathematician averages data collected from a thousand or ten thousand households and learns, for example, that the ‘normal’ male height is five feet and seven inches, that the ‘normal’ time spent reading the newspaper is twelve minutes, that the ‘normal’ diet consists of eggs, potatoes, and meat broth A height of five feet and five inches, or six feet and two inches, is aberrant; spending five minutes, or thirty, with the newspaper is abnormal; too much fish or too few eggs on a plate is deviant Drawing from this data, the mathematician observes regularities: most men are two inches too tall or too squat, most readers linger three minutes too long (or skim three minutes too short) over their newspaper, most housewives cook six potatoes per week too few or too many But it was not only physical or mental traits that a mathematician might average Morality, too, could offer itself up to calculation An analysis of police statistics would reveal the defining characteristics of the ‘average’ criminal, ‘even for such crimes that seem to escape all human foresight like murder since they are committed in general without motive and in circumstances, apparently, the most fortuitous.’ According to Quetelet’s figures, the ‘typical’ murderer would be male, in his twenties, literate and a white-collar worker He would have alcohol on his breath, and be wearing the light clothes of summer Probability would put a pistol (rather than a knife, or a bat, or a vial of poison) in his hand This idea spread quickly It proved popular with scientists and the general public alike Most frequently it was reinforced neither by logic nor analysis, but by simple prejudice People laughed, and sneered, and denounced, and ridiculed variously identified ‘types’ of the average man But worst of all, they believed that such beings really did exist Images propagated the idea with particular efficacy; as the proverb says: a picture is worth ten thousand words One newspaper caricature was enough to wipe out all trace of Quetelet’s wordy disclaimers (his original book had run to several hundred pages) If the caricature depicted an Irishman – all Irishmen – with a misshapen jaw, feathered cap, protruding teeth, it was plain enough that the ‘average Irishman’ bore more than a passing resemblance If it showed a filthy, vulgar, ginswilling beggar, it confirmed how many readers imagined the ‘average poor’ Photography – then still a young and innovative technique – was similarly enlisted Mugshots of eight different individuals were superimposed to reveal the blurry face of the ‘average criminal’ Nine photos of consumptive patients were merged to produce a portrait of tuberculosis Images from six medals depicting Alexander the Great were morphed to reveal the ancient king’s probable features ‘It is now proposed,’ announced one magazine, ‘to get a clear idea of Nebuchadnezzar from the various stone and brick slabs upon which his face is graven.’ But if photography served to popularise the idea of ‘average men’, it also produced an alternative way of looking at ourselves Identikits, created at the end of the nineteenth century, helped to divert some of the focus back from the typical towards the individual Photos that magnified and emphasised all manner of different facial features now replaced the phantom faces typifying this kind of man, or that sort of person Instead of a single abstract representative of this or that contrived category, the images highlighted a wealth of actual noses, foreheads, wrinkles, ears, chins, eyelids, lips and mouths Take the nose, for example Plainly, nobody has a ‘nose’ – a person has a low nose or a high nose, a wide nose or a narrow nose, a hooked nose or a straight nose, a long nose or a snub nose Perhaps the tip is bulbous, the nostrils dilated And what about the chin? Big or small? Flat or bumpy? Does it retreat towards the throat, or jut out proudly? Does it form a square, or slope down into a point? A new picture of the person emerges Yes, of course, commonalities remain His name equally belongs to other people, his nose to other faces We are all made of the same blood and bones But take a closer look See the proportions, the interplay between all the various parts? Every combination, like a mosaic, is unique He has his father’s eyes, and his mother’s curls, and his uncle’s lop-sided smile Together they create something, and someone, new Someone who will look through those eyes in his own way, who will wear that hair according to his own style, who will deploy that smile for his own reasons Talk to that person Watch the skein of laughter lines that diagram his face And how his eyes glisten, or darken, at the sound of certain words He is simply being himself Quetelet (and many others after him) believed that the essence of human nature could be found in the average, but he was mistaken The essence of human nature is its endless variety As Stephen Jay Gould would later remark, ‘All evolutionary biologists know that variation itself is nature’s only irreducible essence Variation is the hard reality, not a set of imperfect measures for a central tendency Means and medians are the abstractions.’ The Cataract of Time If, as is often said, lifetimes flow like a river, they begin with a trickle and culminate in a cataract Heraclitus, the ancient Greek philosopher, put it well when he said, ‘Time is a game played beautifully by children.’ Perhaps this is the root of nostalgia: less the desire to return to our early years, than to the more capacious experience of time that we inhabited as children Time You know how it goes After the age of thirty, I found, the days begin to run away from us We struggle to keep up That is when the nostalgic impulse awakens, burgeons, pesters Last year I moved to Paris Something about being back in a big city after so many years meant I could not help thinking, more and more, about the old London neighbourhood of my youth I had reached that age when the past becomes so big and so deep that your mind finds itself increasingly drawn there It is like living on a fragile coast, by an imposing sea whose smells and sounds gradually overwhelm your senses So I decided to return There was salt water to cross, and buses that jolted and trains that bored, but none of those things mattered I simply had to go and see the place again after all this time My younger siblings, when I spoke to them about the plan, chorused dissuasion ‘There is nothing there,’ they said, perplexed Evidently, they had all moved on ‘Why go back?’ I tried to explain I opened my mouth, felt the breath on my tongue form plausible shapes But my heart was not in it and each of my made-up reasons fell flat I decided not to argue I booked the tickets, packed my bags and left That I could not explain my desire to return to my old London home did not disturb me On the contrary, its surprising strength convinced Reassured, even Gazing out from the wobbly train, I tried to remember when I had last set foot there My reflection in the window wore an expression of thought Tall trees and green hills ran past I looked away, cracked open a book, and stared at the pages till the words seemed to gel into a single inky mass Five years Already? Where had they gone? So many things that I had accomplished, people that I had met, places that I had seen, now looking back, seemed to have taken hardly any time at all And yet how difficult, how exhausting, how important each event had struck me in the moment! And how impossibly distant, a lifetime away, these hours in the train out of Paris would have appeared to me back then It was a fast service to London, without delays Arriving in the capital, I felt more like a suburban commuter than an international traveller I changed trains and rattled out from the centre toward the familiar periphery, my excitement building Gradually the carriage emptied of its suits; a different class of passenger took their place ‘We must be getting close,’ I thought, straining forward, oblivious to my wristwatch Near the end of the line, I gathered my things – my bearings too – and stepped out The platform was covered in litter and broken glass, but for an instant, at least, it felt unambiguously good to be back Time is more than an attitude or a frame of mind It is about more than seeing the hourglass as half empty or half full More than ever in this age, let us call it the computer age, a lifetime has become a discrete and eminently measurable quality To date, to believe the surveys in newspapers, I have spent some one hundred thousand minutes standing in a queue, and five hundred hours making tea I have spent a year’s worth of waking days on the hunt for lost things This year, I knew, contained my twelve thousand and twelfth day and night That number equates to over a quarter of a million hours, seventeen and a quarter million minutes Counting one number for every second since my birth, I had recently made it into the billionaire’s club We occasionally liken time to money, as something to be spent wisely, but it is not money No refunds are possible for days ill-spent; no bank exists to take savings We cannot apportion our time like money, since we live always in ignorance of when the former will end How to plan when a person can never know if he will see tomorrow, or survive to such an age that his eyes turn coalblack with blindness? Perhaps it would be better to talk about time in the manner of certain tribes Strangers to clocks, they pace their days according to nature Native Americans traditionally planted corn ‘when the leaf of the white oak was the size of a mouse’s ear’ Equinoxes and solstices scheduled their rituals As for language, the Sioux have no word for ‘late’ or ‘waiting’ In Australia, the Aborigines believe that time, place and people are one A glance at a tree or a face suffices to know the hour and the day Their discrimination of the seasons is precise, depending on such factors as plant life and changes in the wind: the Eastern Gunwinggu, for example, speak of six seasons – three ‘dry’ and three ‘wet’ – where non-Aborigines see only one of each For these and other tribes, time is the product of our actions It appears when we sing a song, climb a mountain or smoke a pipe, and vanishes when we sleep They not think of time as something pervasive, like the air Seconds, minutes and hours – these are all things that we In place of these terms, they speak of a ‘time of harvesting’ or a ‘river fish time’ Ask an African herdsman how long such-and-such task might take and he replies, ‘Cow milking time’, meaning the time it takes to milk a cow What is an hour to such a man? Perhaps the time it takes to milk ten cows We can put it another way: hour = 10 milkings My equivalent would be hour = 10 tea-makings Let us call it ‘tea time’ A short walk that lasts eighteen minutes equates to three milkings or makingsof-tea; a two-minute advert break amounts to one-third of a cup of tea Between the opening and closing whistles of a football referee, time enough would pass to milk fifteen cows, or make fifteen cuppas I not mean by this digression to suggest that approximations necessarily trump exactitude It is not at all my intention to run clocks down But the particular words and images our respective cultures deploy, shape the way in which we experience time I said just now that time is not money; we might say instead that it is closer to the spending of money According to the tribesmen’s way of thinking, it is what happens when, for example, we enter a marketplace This emphasis on activity in how we think about time strikes me as being very healthy When I hear someone complain about all the hours or weekends he has to fill, I stop and think that it is a mistake to speak of days as we would speak of holes One hole is much the same as any other, whereas every day is different In this, it is more like dough that we can sculpt into infinitely varying shapes On the journey back to my childhood home, I paused outside the train station, then made my way north toward the high street The buildings were more or less the same as in my recollection: the same squat walls, tattooed with graffiti; the same ‘50% off’ signs in shop windows; the same boys and girls, their busy fingers unwrapping sweets No bravado in the architecture, no colour or charm Along the pavements, no bustle either – either too early or too late for shoppers Few cars animated the road I walked mechanically, turning here and there, smelling the sugar of freshly laid tarmac on Waterbeach Road I landed finally on my old street I took it all in On the left stood metal railings and distantly behind them the classroom buildings of my former primary school, factory-long To the right, a chain of brick houses, close set Their thin walls, I recall, made bad neighbours Down the road, I spotted a small man in the distance The man grew bigger with every step He was wearing a blue and red football shirt, but he did not look like a footballer The tightness of the shirt pronounced a sizeable paunch His dark hair was cut penitentiarially short His breathing rasped as he passed me by And then he was gone I was surprised by how little had been altered Painted house number signs, wooden gates, hedgerows all long forgotten, I recognise instantly And yet it all seems so different from my kid days Something has shifted out of sync, something I try to put my finger on In frustration, I walk up and down the street until my legs tire Only as I ready myself for the ride back does it hit me What has changed here is: time In his 1890 classic work, Principles of Psychology, the American philosopher William James noted, ‘The same space of time seems shorter as we grow older – that is, the days, the months, and the years so; whether the hours so is doubtful, and the minutes and seconds to all appearance remain about the same.’ James goes on to cite a mathematical explanation for this phenomenon, by a contemporary French professor According to this professor, Paul Janet, our experience of time is proportional to our age For a ten-year-old child, one year represents one tenth of his existence; whereas for a man of fifty, the same year equates only to one fiftieth (two per cent) The older man’s year will thus seem to elapse five times faster than the child’s; the child’s, five times slower than the man’s What matters, then, is the relationship between one sequence of years and another sequence The interval spanning the ages of thirty-two and sixty-four will seem to the individual of similar duration to that experienced between the ages of sixteen and thirty-two, and to the interval between the ages of eight and sixteen, and to that from the age of four to eight, each having the same ratio For the same reason, all the years from the age of sixty-four to one hundred and twenty-eight (assuming such an age were ever attainable) would seem to us to occupy no more of our feeling, thought, pain, fear, joy and wonder than that big bang epoch between our second and fourth year More recently, from an American called T.L Freeman, we have a formula using Janet’s insight that yields the individual’s ‘effective age’ Freeman’s calculations suggest that we experience a quarter of our entire lifetime by age two, over half by age ten, and more than three-quarters by our thirtieth birthday At only about the chronological midway point, a forty-year-old will experience his remaining time as seemingly but one-sixth of what has gone before For a sixty-year-old, the future will seem to last merely one-sixteenth the duration of his past Are all our attempts to look back, to relive some bygone period, in vain? We can never walk down the same street twice Those streets of my youth belong to another time, which is no longer my own Except, that is, when I dream Fast asleep, I become a visitor there I see a schoolgirl at the edge of the hopscotch grid, contemplating her throw A man, atop a ladder, is washing his windows His free hand glides rhythmically upon the glass On the pavement, a neighbour’s tabby squirms in the sunshine: stretching, and stretching his paws The grunts and sighs of passing traffic fill my ears I see my grandfather, alive, standing with his cane at the gate, as though keeping guard over my father’s vegetable patch I stop and watch my father Sleeves hoisted to his elbows, he picks beans, sows herbs and counts cucumbers I watch without hurry, without a care in the world Time is dilated; there is no time Our body keeps time a great deal better than our brain Hair and nails grow at a predictable rate An intake of breath is never wasted; appetite hardly ever comes late or early Think about animals Ducks and geese need only follow their instinct for when to up sticks and migrate I have read of oxen that carried their burden for precisely the same duration every day No whip could persuade them to continue beyond it We wear the tally of our years on our brow and cheeks I doubt our body could ever lose its count Like the ox, each knows intimately the moment when to stop Higher than Heaven On 22 January 1886 Georg Cantor, who had discovered the existence of an infinite number of infinities, wrote a letter to Cardinal Johannes Franzen of the Vatican Council, defending his ideas against the possible charge of blasphemy A devout believer, the mathematician considered himself a friend of the Church God, he believed, had used his preoccupation with numbers to reveal a further aspect of His infinite nature Fellow logicians had mostly sidestepped the young man’s thinking; hardly anyone yet took seriously the outstanding insights that would make his name Before Cantor it had been impossible to speak mathematically about different sorts of infinity All collections without a final object (the sequence of odd or even numbers, for example, or the primes) were simply conceived as being of equal size Cantor proved that this was false His papers were the first to demonstrate uncountable sets of numbers, that is to say, numerical sequences that even an infinitely long recitation could not exhaust What is more, each uncountable set of numbers spawned another set of numbers that was even ‘bigger’ than the last Of the making of such sets, Cantor realised, there was no end The mathematician Leopold Kronecker, for whom ‘God created the integers [whole numbers], all else is the work of man,’ had no truck with Cantor’s (infinite) tower of ‘smaller’ and ‘bigger’ infinities He hounded his rival with violent words, called him a charlatan, a corrupter of youth In the absence of his peers’ understanding, Cantor turned at last for support to the Holy See The dialogue between theology and mathematics – varied, fitful, and singular – has a long history Above all, infinity became the favourite topic God is infinite, therefore mathematics is religion: a pathway to knowledge of the divine This is what the Church fathers reasoned, and this is why the monks long ago proceeded where the mathematicians had feared to tread A thousand years before Cantor, in an Irish monastery, a man sat day after day at a table smelling of wicks and manuscripts He spent years almost immobile, in deep and sustained contemplation, meditating on a perfect sphere that exists beyond space, universal and without limit Of course, it is contradictory to think about a shape that has no border The monk knew this He knew that to think about infinity is to think in contradictions Minutes passed, hours passed But what is a minute or an hour when compared to eternity? No time at all A minute, an hour, a year, a thousand years are all equally long or short in comparison The light in the monk’s cell would gradually disperse at the end of each long day; his mind might stutter, ‘I, I, I, I, I ’, but try as he would, Johannes Scottus Eriugena – John of Ireland – could not escape his senses and grasp the infinite According to Eriugena, God is not good, since He is beyond goodness; not great since He is beyond greatness; not wise since He is beyond wisdom God, he writes, is more than God, more than time, infinitas omnium infinitatum (the infinity of all infinities), the beginning and end of all things, though He Himself had no beginning and will meet no end Eriugena recalls the words of Job Can you search out the deep things of God? Can you find out the limits of the Almighty? They are higher than heaven — what can you do? Deeper than Sheol — what can you know? Their measure is longer than the earth and broader than the sea If He passes by, imprisons, and gathers to judgment, then who can hinder Him? If God is infinite, Holy Scripture, being inspired by God, is held to exist outside the bonds of conventional time Eriugena cites St Augustine to affirm that the Bible often employs the past tense to express the future Adam’s life in Paradise ‘only began,’ occupying no real time at all, so that its depiction in Genesis ‘must refer rather to the life that would have been his if he had remained obedient’ Augustine’s teachings contributed greatly to the Irish monk’s thought, and that of the theologians who followed In The City of God, Augustine insists that God knows every number to infinity and can count them all instantaneously ‘If everything which is comprehended is defined or made finite by the comprehension of him who knows it, then all infinity is in some ineffable way made finite to God, for it is comprehensible by his knowledge.’ Two centuries after Eriugena, in 1070, Anselm provided his famous ‘ontological proof’ that God is that-than-which-nothing-greater-can-be-thought If every number has its object, the object of infinity is God Anselm became Archbishop of Canterbury; one of his successors, Thomas Bradwardine, in the fourteenth century, identifies the divine being with an infinite vacuum The finite world is compared to a sponge in a boundless sea of space Infinity begets finitude, and thus cannot be grasped in finite terms But how then to understand infinity in infinite terms? Alexander Neckham, a twelfth-century reviver of interest in Anselm’s work, offered this problem a vivid image For Neckham, God’s immensity is such that even if one were to double the world in the next hour, and then triple it in the hour after that, then quadruple it in the following hour, and so on, still the world would be but a ‘quasi point’ in comparison Such immensity inspires in the monks at once admiration and consternation: consternation, because an infinitely remote divine being would rule out the Incarnation For the same reason, the believer would never see God in the Beatific Vision, and neither could he ever conform his will to the divine will The vacuum is in fact a chasm, forever separating Mankind from its Creator The De Veritate of Thomas Aquinas, written between 1256 and 1259, offers a solution: ‘as the ruler is related to the city, so is the pilot to the ship’ An infinitely powerful ruler bears no direct comparison to a humble captain, yet both possess a ‘likeness of proportions’: a finite quantity equates to another finite quantity, in the same way that the infinite is equal to the infinite In other words, ‘three is to six as five million is to ten million’ bears a likeness to the proportion ‘God is to the angels as the infinite vacuum is to an eternal creation’ Aquinas deploys the analogy throughout his work: as our finite understanding grasps finite things so does God’s infinite understanding grasp infinite things; as our finite intellect is to what it knows, so is God’s infinite intellect to the infinitely many things He knows; just as men distribute finite goods so does God distribute all the goods of the universe Aquinas writes that the similarity between the infinite God and His finite creation constitutes a ‘community of analogy The creature possesses no being except insofar as it descends from the first being, nor is it named a being except insofar as it imitates the first being.’ Exasperated by critics he called ‘murmurers’, Aquinas sought to settle a further point of contention The Church taught that the world had a beginning in time ‘The question still arises whether the world could have always existed.’ He penned these words in 1270, entitling them De Aeternitate Mundi (On the Eternal World) His argument was that if the world has always existed, the past regresses infinitely The world’s history must comprise an infinite sequence of past events If there exists an infinite number of yesterdays, then an infinite number of tomorrows must also succeed Time is infinitely past, and infinitely future, but never present For how can any present moment arrive after infinitely many days? Before this potentially unsettling line of reasoning, Aquinas remained unmoved and unimpressed Half hearted were his remonstrations Any past event, like the present moment, is finite: therefore the duration between them is also finite, ‘for the present marks the end of the past’ And what about the succession of past events? Aquinas says the arguments for them can go either way Perhaps God, in all His power, has created a world without end If so, nothing obliged Him to populate it before Mankind A contemporary, Bonaventure, disagreed with Aquinas’s equity His blood thumped at the thought of an interminable past ‘To posit that the world is eternal or eternally produced, while positing likewise that all things have been produced from nothing, is altogether opposed to the truth and reason.’ And what about the contradictions? For instance, if the world were eternal, tomorrow would be a day longer than infinity But how can something be greater than the infinite? In the fourteenth century, Henry of Harclay also faulted Aquinas for saying that an eternal world was possible, but from an entirely opposed point of view to Bonaventure’s For Harclay it was in fact probable, and every supposed contradiction dissolved on careful scrutiny How can something be greater than the infinite? Look, said Harclay, at the infinite number of numbers: we can count from two, or from one hundred, and in both instances never reach a final number, though there are more numbers to count in the first infinity than there are in the second He invoked Aquinas’s proportions to defend the thesis of an infinite universe in which the infinitely many months occur twelve times more frequently than the infinitely many years To those who point out that an infinite past would have produced an infinite number of souls with infinite power like God, Harclay refutes the argument as follows: infinitely many souls would not constitute an infinite power They would be not ‘any species of number, but a multitude of infinitely many numbers.’ Within this endless multitude, every possible number (59, 1,043,962, 999,999,999,999,999,999,999,999,999,999 ) could be found, distinct and finite, each corresponding to a soul; save, that is, for an infinitieth number/soul since this would produce a contradiction: ‘there is not a number of infinite numbers, for then it would contain itself, which is impossible.’ We trace to the same period, in the monk Gregory of Rimini’s hand, the first definition of an infinite number as that which has parts equally great as the whole: an infinite sequence can be part of another infinite sequence and is equal to the infinite of which it is a part Every twenty-third number for example (we might just as well have taken every ninety-ninth number, or every third, or every five billionth) in the infinite succession of counting numbers (1, 2, 3, 4, 5, ) produces a sequence as long – infinitely long – as all the counting numbers combined: match one with twenty-three, two with forty-six, three with sixty-nine, four with ninety-two, five with one hundred and fifteen, and so on, ad infinitum Gregory articulated his defining idea fully five centuries before Cantor He taught for many years in Paris, at the Sorbonne, where his pupils called him Lucerna splendens Perhaps in him they sensed, as future scholars would claim, the last great scholastic theologian to wrestle with the infinite John Murdoch, a historian of mathematics at Harvard University, remarked that Gregory’s insight received hardly any notice from his peers or successors Since the ‘equality’ of an infinite whole with one or more of its parts is one of the most challenging, and as we now realise, most crucial aspects of the infinite, the failure to absorb and refine Gregory’s contentions stopped other medieval thinkers short of the hitherto unprecedented comprehension of the mathematics of infinity which easily could have been theirs In his writings, Cantor described himself as a servant of God and the Church His ideas had struck him with the force of revelation It had been with God’s help, he said, that he had worked day after day, alone, at his mathematics But the mathematician was far from angelic; his humility sometimes slipped To a friend, in 1896, Cantor confided in an excess of pride ‘From me, Christian Philosophy will be offered for the first time the true theory of the infinite.’ The Art of Maths I met a mathematician at a ‘conference of ideas’ in Mexico at which we had both been invited to speak He was from the United States, and like all the mathematicians that I have ever crossed in my travels he fell immediately to talking shop Moving to a corner of the conference green room, he talked to me about the history of numbers in Cambodia The concept of zero, he believed fervently, the familiar symbol of nothingness, hailed from there He dreamed of trekking the kingdom’s dirt tracks, in pursuit of any surviving trace More than a millennia separated him from the decimal system’s creation; the odds of turning up any new evidence were slim But he did not mind He began to explain his current research in number theory, talking quickly with the compression of passion, and I listened intently and tried to understand When I understood, I nodded, and when I did not understand, I nodded twice, as if to encourage him to move on His words came fast and enthusiastic, opening up vistas that I could not quite see and mental regions into which I could not follow, but still I listened and nodded and enjoyed the experience very much Occasionally I supplemented his ideas and observations with some of mine, which he received with the utmost hospitality It always feels exciting to me, the camaraderie of conversation: no matter whether it involves words or numbers He had none of the strange tics or quirks of the mathematicians that we find in books or see in movies From experience, I was not the least surprised Middle-aged, he looked fit and slender, though with skin as pasty as a writer’s His shirt was open at the neck His face wore many laughter lines When our time was up, too soon, he patted his pockets and withdrew from one of them a small notebook in which he habitually jotted down his random thoughts and sudden illuminations As he wrote out his contact information for me, I noticed the smallness and smoothness of his hands ‘Great meeting you.’ We promised to stay in touch It was still a pleasant surprise, coming down next morning to the hotel restaurant for an early breakfast, to hear the mathematician’s voice call me over to his family’s table I passed the assorted reporters munching their bowls of cereal, and various conference ‘stars’, dodging coffee-flecked waiters and pushing empty chairs out of my way, until I reached them The mathematician smiled at his wife (also a mathematician, I learnt) and the surprisingly placid teenage girl sat in between who looked a lot like her mother Their flight out was still a few hours away: over tea and toast, we talked We talked about the Four Colour Theorem, which states that all possible maps can be coloured in such a way that no district or country touches another of the same colour – using only (for instance) red, blue, green, and yellow ‘At first sight it seems likely that the more complicated the map, the more colours will be required,’ writes Robin Wilson in his popular account of the puzzle’s history, Four Colours Suffice, ‘but surprisingly this is not so.’ Redrawing a country’s boundary lines, or imagining wholly alternative continent shapes, makes no difference whatsoever One aspect of the problem, in particular, had long intrigued me After more than a century of fruitless endeavours to demonstrate the theorem conclusively, in 1976 a pair of mathematicians in the United States finally came up with a proof Their solution, however, proved controversial because it relied in part on the calculations of a computer Quite a few mathematicians refused to accept it: computers cannot maths! ‘I actually met one of those guys who came up with the proof,’ my new friend recalled, ‘and we discussed how they had found just the right way to feed the data into the machine and get an answer back It really was a smart result.’ What did he and his wife think of the computer’s role in mathematics? In answer to this general question they were more circumspect The Four Colour Theorem’s proof, they admitted, was inelegant No new ideas had been stimulated by its publication Worse, its pages were almost unreadable It lacked the intuitive unity, and beauty, of a great proof Beauty How often have I heard mathematicians employ this word! The best proofs, they tell me, possess ‘style’ One can often surmise who authored the pages simply from the distinctive way that they were put together: the selection, organisation and interplay of ideas are as personal, and as particular, as a signature And how much time might they spend on polishing their proofs Superfluous expressions, out! Ambiguous terms, out! Yes, but it was worth all the trouble: well-written proofs could become ‘classics’ – to be read and enjoyed by future generations of mathematicians ‘What time is it?’ None of us was wearing a watch We stopped a waiter and asked ‘Already?’ said the mathematician’s wife when she heard his answer They drained their cups, and dispersed their crumbs, and made shuffling sounds with their feet ‘Oh,’ said the mathematician, turning back to me, ‘I forgot: where did you say you were based again?’ What with the history of the decimals, and the winding numerical vistas, and the painting of the entire globe with the colours of a single flag, the accidental features of our lives – where we lived, with whom, under what roof and colour of sky – had been completely absent from our conversations I told him ‘Paris,’ he echoed ‘Why, we love Paris!’ France’s capital has something of a one-sided reputation as the consummate city of artists We know it as the city of Manet, of Rodin, of Berlioz; as the city of street singers and can-can dancers; as the city of Victor Hugo and of young Hemingway in A Moveable Feast: scribbling in a café corner, turning coffee and rum and the strictures of Gertrude Stein into stories But Paris is also the city of mathematicians Its researchers, a thousand strong, make the Fondation Sciences Mathématiques de Paris (FSMP) the largest group of mathematicians in the world About one hundred of the city’s streets, squares and boulevards are named after their predecessors In the twentieth arrondissement, for example, one can walk the length of rue Evariste Galois, named after a nineteenth-century algebraist felled at the age of twenty by a dueller’s bullet On the opposite side of the Seine, in the fourteenth arrondissement, lies rue Sophie Germain whose namesake introduced important ideas in the fields of prime numbers, acoustics and elasticity before her death in 1831 According to her biographer Louis Bucciarelli, ‘She did not wish to meet others in the streets or houses of the day, but in the purer realm of ideas outside time, where person was indistinguishable from mind and distinctions depended only on qualities of intellect.’ A few minutes’ walk away is Fermat’s little road There are also streets called Euler, and Leibniz, and Newton Among the letters waiting for me on my return to my adopted home was one from the city’s Fondation Cartier A museum for contemporary art, it had sent me a preview invitation to its upcoming exhibition ‘Mathematics: A Beautiful Elsewhere’: the first in Europe to showcase the work of major living mathematicians in collaboration with world-class artists The timing seemed doubly auspicious: October 2011 happened to be the two-hundredth anniversary of Galois’s birth The museum stands in the fourteenth arrondissement at the lower end of one of the long boulevards that diagram the city It is an ostentatiously modern building, all shiny glass and geometric steel, bright and spacious, an example of ‘dematerialised’ architecture Reflected in the glass, scraggly trees denuded of their summer foliage appeared twice I looked up at the symmetrical branches as I passed and entered Mathematics and contemporary art may seem to make an odd pair Many people think of mathematics as something akin to pure logic, cold reckoning, soulless computation But as the mathematician and educator Paul Lockhart has put it, ‘There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.’ The chilly analogies win out, Lockhart argues, because mathematics is misrepresented in our schools, with curricula that often favour dry, technical and repetitive tasks over any emphasis on the ‘private, personal experience of being a struggling artist.’ It was the mathematicians’ artistic impulse, and inner struggle, that the exhibition’s organisers intended both to communicate and celebrate A white interior, zero-shaped, was the work of the American filmmaker David Lynch Walls usually reserved for frames and canvases lent their space to equations, light effects and number displays I walked through the rooms, now bare and silent, now colourful and stimulating, stopping here and there to take a closer look I watched the other guests stand back and point and converse in low voices Before a bright collage of sunrays and leopard spots, waves and peacock tails, and the underlying equations for each, fingers swayed and eyes widened Another hall arrested visitors’ feet around a lean aluminium sculpture, its curves reaching toward infinity But, for me, the highlight of the exhibition took place in a darkened room downstairs Here the visitors melted into twilight, rendered homogeneous in the darkness, sitting or standing in silence, all eyes, observing a large screen where a film shot in black and white was playing A youngish face, screen big, was talking about his life as a mathematician I pressed my back against the far wall and listened as he spoke of ‘fat triangles’ and ‘lazy gases’ Three or four minutes old, the film suddenly altered: the face gave way to another, wearing glasses Four minutes after this, the face changed again: this time, a woman’s began to speak about chance In total, the film lasted thirty-two minutes – eight faces long The men and women featured came from a wide range of mathematical subdisciplines – number theory, algebraic geometry, topology, probability – and spoke either in French, or English, or Russian (with subtitles), but their passion and wonder linked each personal testimony into a fascinating and involving whole Two of the testimonies, in particular, stood out They reminded me of my conversations with the mathematicians in Mexico, and with those in other lands, and the feelings of kinship and excitement that these exchanges incited within me During his four minutes, Alain Connes, a professor at the Institut des Hautes Études Scientifiques, described reality as being far more ‘subtle’ than materialism would suggest To understand our world we require analogy – the quintessentially human ability to make connections (‘reflections’ he called them, or ‘correspondences’) between disparate things The mathematician takes ideas that are valid in one area and ‘transplants’ them into another hoping that they will take, and not be rejected by the recipient domain The creator of ‘noncommutative geometry’, Connes himself has applied geometrical ideas to quantum mechanics Metaphors, he argued, are the essence of mathematical thought Sir Michael Atiyah, a former director of the Isaac Newton Institute for Mathematical Sciences in Cambridge, used his four minutes to speak about mathematical ideas ‘like visions, pictures before the eyes.’ As if painting a picture or dreaming up a scene in a novel, the mathematician creates and explores these visions using intuition and imagination Atiyah’s voice, soft and earnest, made attentive listeners of everyone in the room Not a single cough or whisper intervened Truth, he continued, is a goal of mathematics, though it can only ever be grasped partially, whereas beauty is immediate and personal and certain ‘Beauty puts us on the right path.’ The faces, old and young, smooth and hairy, square and oval, each had their say Gradually, the room began to empty Its intimate ambience slowly dissolved I followed the last group of visitors up the stairs and out the building and not a word was exchanged The night absorbed us I walked for a while, beside the river, with the night in my hair and in my pockets and on my clothes The night, I know, is tender to the imagination; at this hour, throughout the city, artists sharpen pencils and dip brushes and tune guitars Others, with their theorems and equations, revel just as much in the world’s possibilities The world needs artists Into words and pictures, notes and numbers, each transforms their portion of the night A mathematician at his bureau glimpses something hitherto invisible He is about to turn darkness into light ... and how we make meaning in our lives Tammet was born in London in 1979, the eldest of nine children He lives in Paris Thinking in Numbers Daniel Tammet www.hodder.co.uk First published in Great... finally brings the gooey avalanche to a belated halt The Brothers Grimm introduced me to the mystery of infinity How could so much porridge emerge from so small a pot? It got me thinking A single... how to count like a Viking Icelanders, I learnt, have highly refined discrimination for the smallest quantities ‘Four’ sheep differ in kind from ‘four’, the abstract counting word No farmer in